Structured Finite Element Methods (FEMs) based on low-rank approximation in the form of the so-called Quantized Tensor Train (QTT) decomposition (QTT-FEM) have been proposed and extensively studied in the case of elliptic equations. In this work, we design a QTT-FE method for time-domain acoustic wave equations, combining stable low-rank approximation in space with a suitable conservative discretization in time. For the acoustic wave equation with a homogeneous source term in a single space dimension as a model problem, we consider its reformulation as a first-order system in time. In space, we employ a low-rank QTT-FEM discretization based on continuous piecewise linear finite elements corresponding to uniformly refined nested meshes. Time integration is performed using symplectic high-order Gauss-Legendre Runge-Kutta methods. In our numerical experiments, we investigate the energy conservation and exponential convergence of the proposed method.

翻译：基于低秩近似（具体形式为所谓的量化张量列分解）的结构化有限元方法（QTT-FEM）已在椭圆型方程情形下被提出并得到广泛研究。本文针对时域声波方程设计了一种QTT-FE方法，将空间中的稳定低秩近似与时间上合适的保守离散化相结合。以单空间维度下具有齐次源项的声波方程作为模型问题，我们考虑将其重写为时间上的一阶系统。在空间上，我们采用基于连续分段线性有限元的低秩QTT-FEM离散化，该有限元对应于均匀细化的嵌套网格。时间积分采用辛高阶高斯-勒让德龙格-库塔方法进行。在数值实验中，我们研究了所提方法的能量守恒性和指数收敛性。